1. Field of Invention
This invention relates to Recursive Parameter Estimators (System Identifiers), such as are used in the engineering fields of Adaptive Control Systems and Signal Processors and in the operations research, econometrics, biomedical signal processing, sonar/radar/optical data processing and resource allocation fields of Time Series Analysis and Prediction.
2. Description of Prior Art
A typical example of the use of a Parameter Estimator or System Identifier is depicted in FIG. 1. A signal process or dynamical system of an assumed known structure is given, but the set (or "vector") of parameters .theta. which specifically define the particular process or dynamical system at hand are assumed not to be known. The objective of a parameter estimator or identifier is to process the measurements to produce a reliable estimate .theta. of the unknown parameters .theta. in a timely fashion. In the class of Least Square Estimators (LSE), which goes back to Gauss, the estimates .theta.(t) are produced by an algorithm that is based on minimization of the sum of the squares of the estimate errors .theta..sub.i -.theta..sub.i , (i=1, 2, 3, . . . , p).
The Parameter Estimator or Identifier is a special-purpose data-processing system or computer, such as a digital computer, analog computer, neural network, or the like. The Estimator/Identifier accepts as inputs two multichannel (multi-dimensional or "matrix-vector") signals: a main (vector) signal y which carries the desired information about .theta., and a secondary (matrix) carrier/regressor references signal (which may be implicit or omitted if its characteristics are known [e.g. a sine wave of known frequency and phase] and it is reconstructed or regenerated as part of the Estimation/Identification process). Under the legal Doctrine of Equivalents, a general purpose computer programmed to execute the algorithm which operates upon the signals obtained from a preprocessing filter which operates upon the input signals and output signals of the process or dynamical system to be identified and produces reliable parameter estimates .theta., can also be regarded as such a special-purpose computer. Both the computer (an apparatus or system) and the procedure or method (a process) of programming and utilizing it constitute Statutory Patentable Subject Matter, as in, e.g. U.S. Pat. No. 4,646,256 (Feb. 24, 1987) on the DBT (Discrete Bracewell Transform).
The prior art in this field is described in such books on Adaptive Control System Design as those by Astrom and Wittenmark [1], Middleton & Goodwin [2], Sastry & Bodson [3], and Slotine & Li [4].
It is not feasible to implement an industrially or commercially useful Parameter Estimator/Identifier except by means of a computing machine or physically implemented data-processing system, whether it be mechanical (pneumatic analog, hydraulic analog), optical (analog or digital), or electrical (analog) or electronic (digital) or hybrid. However, the architectural item which distinguishes one class of Estimator/Identifiers from another is the specific data processing Algorithm (or analog equivalent) which defines the information-processing architecture and the method of implementing and using the Estimator/Identifier.
As documented in the literature [1]-[4], Parameter Estimator/Identifiers are in widespread industrial and commercial use, and constitute Statutory Patentable Subject Matter. However, the disclosure of an invention in the category of a novel, useful and non-obvious Estimator/Identifier incorporating a novel, useful and non-obvious Algorithm does not preclude the bare Algorithm itself from being used in other non-industrial and non-commercial ways apart from the novel Estimator/Identifier (e.g. by hand-computation of an illustrative or academic numerical example).
Nevertheless, what chiefly distinguishes one Parameter Estimator/Identifier from another is the performance characteristics of the Estimation/Identification Algorithm which the Estimator/Identifier incorporates and is constructed or programmed to implement.
The principal performance characteristic is "convergence": does the set of parameter estimates .theta.(t) eventually stop changing (or exhibit only tolerably small residual fluctuations) as time t increases, and do the estimate errors (.theta.-.theta.) decrease to acceptability? A Parameter Estimator/Identifier which fails to converge has no utility.
The second performance characteristic is "quality of convergence"; there are many ways of defining quality of convergence but in the prior art the traditional criterion is based on the monotone non-increasing behavior of the estimate errors obtained by minimizing the integral (or sum) of the square of the estimate errors. This is called Least Squares Estimation (LSE). However, the present invention discloses a new quality of convergence category, Zero-Error Equilibrium (ZEE), which is superior to LSE as demonstrated herein.
The third and final performance characteristic is "speed of convergence". The more rapidly the Estimator/Identifier converges to the desired estimate, the more useful and more economical the Estimator/Identifier will be.
The prior art Estimator/Identifiers all suffer from two severe shortcomings with respect to the first and third performance characteristics. Specifically, for a reason to be clearly disclosed and explained herein, there is a large class of operating conditions and operating environments wherein the prior art Estimator/Identifiers fail to converge, resulting in a catastrophic failure of the System Identifier, which, if it is a subsystem of an Adaptive Control System, can result in a catastrophic failure of the entire system. For example, several early experimental Adaptive Flight Control Systems have suffered catastrophic failures, resulting in test-pilot fatalities.
Accordingly there has been a need for improved Estimators/Identifiers of guaranteed ID convergence characteristics (where the acronym ID is used as a common abbreviation for "identification").
In some classes of Parameter Estimator/Identifiers (e.g. those based upon Extended Kalman Filtering, denoted by EKF) there is a possibility of convergence to a parameter estimate which is quite different from the true value of the parameter. This is caused by the algorithm achieving a local but not a global minimum with respect to the criterion of Minimal Variance Estimation (MVE). However, in the presently disclosed art, hereafter called Zero-Error Equilibrium (ZEE), the structure of the algorithm and the class of potential applications is so chosen that this problem does not arise, i.e. the ZEE algorithm has only one equilibrium state and that corresponds to zero steady-state "ID estimate error". The convergence of the ZEE algorithm to any estimate automatically guarantees convergence to an acceptable estimate.
All prior known Parameter Estimators/Identifiers also suffer from susceptibility to very sluggish convergence; there are very common, specific conditions disclosed herein under which such Estimator/Identifiers converge so sluggishly as to vitiate any potential economic or practical utility.
Accordingly there has been a need for Estimator/Identifiers of prespecifiable guaranteed rates of convergence, particularly exponential convergence, if achievable.
In conclusion, there has been a need for Parameter Estimator/Identifiers of guaranteed convergence to zero error, at prespecifiable guaranteed rates.